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1. ## American Options-early exercise

I dont understand the solution to this problem:

You are given the following information at time-0:
(i) The price of a stock is \$99.
(ii) The stock pays 3 discrete dividends: \$1.00 in 3 months, \$1.50 in 6 months, and \$2.00 in 9 months.
(iii) The continuously compounded risk-free interest rate is 8%.
At time-0, a 1-year American call option was written on the stock. The call option has a strike price of 100. The call option did not expire worthless.
Determine when the call was exercised.
Time 0
Time 3 months
Time 6 months
Time 9 months
Time 12 months

I know you exercise early when the PV(Div) > K(1-exp^(-r(T-t)))

The solution has
at t=.25 the PV(Div) is 1 +1.5exp^(-.08*.25) +2exp^(-.08*.5)

why isnt the PV(Div) at t=.25 just 1exp^(-08*.25)??

2. Originally Posted by jorgeb1987
I dont understand the solution to this problem:

You are given the following information at time-0:
(i) The price of a stock is \$99.
(ii) The stock pays 3 discrete dividends: \$1.00 in 3 months, \$1.50 in 6 months, and \$2.00 in 9 months.
(iii) The continuously compounded risk-free interest rate is 8%.
At time-0, a 1-year American call option was written on the stock. The call option has a strike price of 100. The call option did not expire worthless.
Determine when the call was exercised.
Time 0
Time 3 months
Time 6 months
Time 9 months
Time 12 months

I know you exercise early when the PV(Div) > K(1-exp^(-r(T-t)))

The solution has
at t=.25 the PV(Div) is 1 +1.5exp^(-.08*.25) +2exp^(-.08*.5)

why isnt the PV(Div) at t=.25 just 1exp^(-08*.25)??
********

The present value of the dividends includes all dividends paid before the call expiration. In this case there is a dividend paid at 3 different times.

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